In the beginning of the game, I came to the town and tried stealing from, pickpocketing, and killing everyone. Now, I have to go back there for a quest called The Horn of Jurgan Windcaller. The girl at the inn won't stop attacking long enough for me to talk to her. What can I do?
As far as I can tell, you have three options.
First, have you tried simply yielding to Delphine (the inn girl)? Sheath your weapon and see if she calms down.
Paying for your crimes
If she's too aggressive to accept the yield, you'll likely need to pay for your crimes. Riverwood's hold (basically its governing city) is Whiterun, so any crimes you committed in Riverwood can be cleared there.
So, head to Whiterun. If the guards attack you on sight, you'll need to yield and see if they accept. Otherwise, simply speak to them and they'll probably recognize you. Pay for your crimes with gold or jail time and head back to Riverwood. If she still attacks you, try yielding again and hoping for the best.
Fix it with the console
If, after all this, she still hates your guts, you can always try using the console (assuming you're on PC) to either clear her aggression or force yourself later in the quest. I've not done this personally so I can't say how effective it will be, but here's the Elder Scrolls Wiki page on the console commands:
I'd recommend only using the console as a last resort, and keeping a backup save as the quest commands can trigger bugs.